Local moment problem

Abstract

The work is devoted to the local moment problem, which consists in finding of non-decreasing functions on the real axis having given first 2n + 1, n ≥ 0, power moments on the whole axis and also 2m + 1 first power moments on a certain finite axis interval. Considering the local moment problem as a combination of the Hausdorff and Hamburger truncated moment problems we obtain the conditions of its solvability and describe the class of its solutions with minimal number of growth points if the problem is solvable.Attempts to extract useful information on correlation functions of real systems from available experimental data and some known exact relations have led us to the following version of moment problem Given two sets of numbers a 0 , ..., a 2n , b 0 , ..., b 2m and an interval [0, Λ], 0 < Λ < ∞. To find a set of non-decreasing measures dσ(t) , which satisfy the conditions:(1)The formulated local moment problem is a special combination of the well known truncated Hausdorff (no conditions a) in (1) and sought σ(t) are constants on (−∞, Λ) and (Λ, ∞) ) and Hamburger (no conditions b) in (1) moment problems [1,2]. A possible approach to the solution of the local moment problem consists in the representation of the sought measure dσ(t) as the sum dσ(t) = dσ Λ (t) + dσ ⊥ Λ (t), where the measure dσ Λ (t) is concentrated on the segment [0, Λ], while the function σ ⊥ Λ (t) has no growth points on [0, Λ]. The retrieval of dσ Λ (t) is reduced to the Hausdorff problem on the interval [0, Λ] for the given set of moments b 0 , ..., b 2m . The quest of dσ ⊥ Λ (t) consists in the search of some special solutions d σ(t) of the truncated Hamburger moment problem, which satisfy the additional restriction σ(Λ − 0) − σ(+0) = 0 for the set of moments(2)We call the latter moment problem the truncated Hamburger moment problem with the gap [0, Λ]. Using as in [3][4][5][6] the approach based on the extension theory of Hermitian operators we obtain first the solvability criterium of the truncated Hausdorff problem, which is treated as a version of the Hamburger problem [1], where the sought σ(t) should be constant out of (0, Λ).Theorem 0.1 A system of real numbers b 0 , ..., b 2m are power moments of non-negative measure dσ Λ (t) on [0, Λ] if and only if a) the Hankel matrix Γ m := (b k+j ) m k,j=0 is non-negative; b) for any set of complex numbers ξ 0 , . . . , ξ r , 0 ≤ r ≤ m − 1, the condition r j,k=0 b j+k ξ k ξ j = 0 (3) implies r j,k=0 b j+k+2 ξ k ξ j = 0; (4) c) the Hankel matrix Γ (1) m−1 := (b k+j+1 ) m−1 k,j=0 is non-negative and for any set ξ 0 , ..., ξ r ∈ C, 0 ≤ r ≤ m − 1, the condition r j,k=0b j+k+1 ξ k ξ j = 0 (5)

Dynamic properties and the roton mode attenuation in liquid 3 He: ab initio study within the self-consistent method of moments

“…Contrary to Stieltjes, Hausdorff [40], and the physically motivated 'local' [43] problems, the distribution density (the spectral density) is extended in the Hamburger problem to the whole real axis of frequencies so that if we select a symmetric spectral density the working formulae simplify significantly (see, nevertheless [44]).…”

Dynamic properties and the roton mode attenuation in the liquid 3He: an ab initio study within the self-consistent method of moments

“…Contrary to the Stieltjes, Hausdorff [38], and the physically motivated "local" [41] problems, the distribution density (the spectral density) is extended in the Hamburger problem to the whole real axis of frequencies so that if we select a symmetric spectral density the working formulas simplify significantly (see, nevertheless [42]).…”